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Middle/High School Algebra, Geometry, and Statistics (AGS)
9.21 Properties of Arithmetic Mean

Some properties of Arithmetic mean
  • If x 1 and x2 are the means of the two groups computed from the values n1 and n2 then the mean x is given by the formula
    x = n1x1+n2x2/ n1+n2

  • If each observation in the data is replaced by x, the sum total of all the observations remains unchanged.
    x = x1,x2,x3,x4........,xn/n
    So x1,x2,x3,x4........,xn = nx
    Replacing each observation by x, we get

    x+x+x........+x = nx

  • If every value of the variable x is either increased, decreased, divided or multiplied by a constant, the observations so obtained also increases, decreases, gets multiplied or gets divided respectively by the same constant.

  • Algebraic sum of the deviation of a set of values from their arithmetic mean is 0
  • Find the sum of the deviations of the variate values 3,4,6,8,14 from their mean.

    Mean of 3,4,6,8,14 is

    Hence the sum of the deviations about mean = 0

    Merits of arithmetic mean
    1.) It is rigidly defined.
    2.) It is based on all the values.
    3.) It is more stable than any other average.
    Demerits of arithmetic mean
    1.) It is highly affected by abnormal values.
    2.) The loss of even a single observation makes it impossible to compute the aritmetic mean correctly.

Q 1: The mean of 13 observations is 14. If the mean of the first seven observations is 12 and of the last seven observations is 16, find the seventh observation.

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